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October 21st, 2013, 07:43 AM  #1 
Newbie Joined: Jun 2012 Posts: 5 Thanks: 0  Approach reals by 3smooth numbers
Hello everyone, I am not a mathematician but I am looking for a theorem. I want to know if it is possible to approximate any real number by the quotient of two 3smooth numbers. In practice that means: can any real number be approached arbitrary close by where m and n are in ? Thanks in advance!! Maurice 
October 21st, 2013, 08:03 AM  #2  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Approach reals by 3smooth numbers Quote:
 
October 21st, 2013, 10:15 AM  #3 
Newbie Joined: Jun 2012 Posts: 5 Thanks: 0  Re: Approach reals by 3smooth numbers
Thanks for your reply. I doubt it also that every real number can be approached that way, but I really like to know how dense the set is and what kind of set is created by and m,n , maybe a kind of Cantor set. All info is welcome. Maurice 
October 21st, 2013, 12:57 PM  #4 
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Approach reals by 3smooth numbers
Well, this isn't a proof, but I believe you can approximate any real number arbitrarily closely with because you can get arbitrarily close to 1. is irrational, and hence you can approximate it arbitrarily closely with rational numbers, which translate into the values of m and n. The closer the ratio m:n is to , the closer and are to 1. Remember, you can approximate arbitrarily closely with rational numbers. Now suppose you want to approximate a real number x which is larger than 1. Take an approximation which is sufficiently close to 1 and greater than 1, and then continually multiply it by itself until you get a number which is larger than x, and call the exponent p. Then will be sufficiently close to x. If the real number x you wish to approximate is less than 1, then simply choose an approximation which is sufficiently close to 1 and less than 1, and continually multiply it by itself until you get a number which is less than x. In particular, the margin of error (percentage wise) in this calculation will be less than or equal to , which can be arbitrarily close to 0.

October 21st, 2013, 01:16 PM  #5 
Senior Member Joined: Sep 2008 Posts: 150 Thanks: 5  Re: Approach reals by 3smooth numbers
Well obviously the numbers of the form are all positive and can thus only be dense in the nonnegative real numbers, but that is true. And in any case the full statement on 3smooth numbers is true. Unfortunately, i don't know of any bigger theorem implying it, so here's the outline of the proof: (It is rather sketchy, but if required I can fill the gaps later) Step 1: It is enough to show: for any there are integral m, n such that . In fact, assuming the assertion: For every real r > 1 and every take f = 2^m 3^m to be a number as described in the assertion, i.e. 1 < f < 1 + e. Then there is an integer k such that . Letting epsilon go to 0 we can thus approximate r. The proof for 0< r < 1 is similar. Step 2: It is enough to show: for any there are integral m, n such that . To prove the assertion of step 1 for any given assume the assertion of step 2 is valid . Then the same pair (m,n) will satisfy the conditions of step 1. Step 3: The assertion of step 2 is well known to be true for any two nonzero reals a,b in stead of log( 2) and log (3) such that the quotient a/b is irrational. There is a nice proof using topological group theory, but i don't know a reference at the moment. maybe someone else knows one. 
October 21st, 2013, 04:04 PM  #6 
Senior Member Joined: Nov 2011 Posts: 595 Thanks: 16  Re: Approach reals by 3smooth numbers
Hi, Yes any number can be approached in this way. As Peter as stated this is equivalent to ask if is dense in R. Note that this space forms a group, which is a subgroup of R. It is easy to show that any subgroups of R are either of the form with a is a real number or are dense in R. Obviously there is no such a in our case, the group has no smallest elements therefor this group is dense in R and any real number can be approached. 
October 21st, 2013, 11:12 PM  #7  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Approach reals by 3smooth numbers Quote:
So, definitely, nice thinking and nice proof, Peter.  
October 22nd, 2013, 06:00 AM  #8 
Newbie Joined: Jun 2012 Posts: 5 Thanks: 0  Re: Approach reals by 3smooth numbers
Thank you all for the replies! It helped me a lot. Is it possible to say now that every positive real can be approached by with m and n in since it can be very close to one. Thanks again! 
October 22nd, 2013, 07:31 AM  #9  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: Approach reals by 3smooth numbers Quote:
 
October 22nd, 2013, 01:34 PM  #10  
Senior Member Joined: Feb 2012 Posts: 628 Thanks: 1  Re: Approach reals by 3smooth numbers Quote:
 

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